Impressive. Half the breakthroughs involve Chomsky.
His politics may motivate people to disregard him, but mostly he's just come to the devastating conclusion (for solid reasons) that people working on language outside the Chomskyan tradition are not doing science; semantics, linguistic anthropology, and some areas of psychology are just wind. People who do such work will not understand his criticism because their academic standing depends on their not understanding it. There is thus a supply of motivated PhDs ready to dismiss and de-emphasize his contributions. Some of the same applies to philosophers who work on language.
Oh, and at last look he was not just the most cited linguist. He was the most cited living human being.
(And FWIW, I came to linguistics from philosophy via formal semantics; I once held negative opinions of Chomsky myself. It took time to see that he was right.)
The most successful machine translation systems don't use Chomsky's grammars. But their compiler's parser probably does. More:
"I found the mathematical approach to grammar immediately appealing---so much so, in fact, that I must admit to taking a copy of Noam Chomsky's Syntactic Structures along with me on my honeymoon in 1961." - Knuth
You've made a very strong claim about linguistic theory while knowing very little about it. The vast majority of those working on phonology do so within the mold set by Chomsky & Morris Halle in 1970, and there are roughly zero syntacticians doing serious work on natural language syntax outside the Chomskyan tradition (whether in GB or Minimalism).
"The religion built around Chomsky is a bit crazy."
You're hardly in a position to judge, with no more than a passing acquaintance with psych 101 summaries of his work.
"Just because you defined something doesn't mean it's impressive."
The Chomsky hierarchy was not "defined" in the deflationary sense that you're pushing. Chomsky discovered that generative mechanisms partake of a special set of related properties relating the range of their output to specific constraints on the formalisms that can exactly describe their grammars. It so happens that this discovery made possible much that happened in CS during the 60s and into the 70s.
I'll leave aside your bizarre comments as to the unimpressiveness of Peano arithmetic.
My point is I think it's a fascinating and historically important fact that Neuro-lingustic programming grew directly out of Chomsky-informed analysis of the communications between therapists and their clients.
If you read "Structure of Magic, Vol. I" it's basically using Transformational Grammar to detect missing information in the surface structure of clients' statements in a therapeutic context.
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I've got to say that that wikipedia article is one of the very few that I have encountered that badly misrepresents its subject. I have direct experience of NLP. I would describe it as the operating system of the human mind.
I don't really know what to make of the idea that it "has been discredited as a pseudoscience by experts" except that they must not have been very good experts.
The Wikipedia article actually does a fairly good job of listing the evidence for this claim (further down). In particular, they show polls of the lack of acceptance by experts within the field of psychology. So the best you could say is that it’s a fringe theory. Furthermore, several claims of the original creators very directly contradict established biological facts (treating physiological ailments such as near-sightedness or the common cold). This obviously casts suspicion on the rest of the theory.